using 10 iterations of the conjugate gradient method.
In what follows, the forward operator \( A \)  assumes a parallel-beam geometry with \( n_d = {362} \)  detectors of size 1 pixel and is discretized using the ASTRA toolbox [1]}.
Unless stated otherwise, \( \eta \)  is \( {0.1}{} \)  Gaussian noise.
[t]
InputInput
OutputOutput

\( \phi \)  approximately minimizing (REF )
\( \mathcal {B} \leftarrow \lbrace  u_1,\cdots ,u_{n_\text{buffer}} \rbrace  \) , \( u_i \sim \mathcal {U}({[0, 1]}^{n}) \) 
\( t = 1,\cdots ,n_e\) 
\( x^+ \sim (p_x * \mathcal {N}(0, \sigma _{\text{data}}^2\mathrm {Id}_n)), x^0 \sim \mathcal {B} \) 
Generate \( x^- \)  with (REF ) using \( x^0 \) , \( \epsilon \) , \( K \) , \( \beta \) 
\( r > p_{\text{re}} \) 
\( x_{\text{refill}} = x^- \) 


\( r > 0.5 \) 
\( x_{\text{refill}} = x^+ \) 

\( x_{\text{refill}} = u \sim \mathcal {U}({[0, 1]}^{n}) \) 

\( \mathcal {B} \leftarrow \mathcal {B} \setminus \lbrace  x^0 \rbrace  \cup \lbrace  x_{\text{refill}}\rbrace  \) 
\( \phi \leftarrow \operatorname{Adam}(\nabla _{2} R(x^+, \phi ) - \nabla _2 R(x^-, \phi )) \) 

Maximum Likelihood training of an ebm.
\( \mathcal {U}(\mathcal {X}) \)  denotes the uniform distribution on \( \mathcal {X} \)  and each \( r \)  denotes an independent sample from \( \mathcal {U}([0, 1]) \) .

[t]
InputInput
OutputOutput
Dodowhile
Breakbreak

\( x^{J + 1} \)  approximately minimizing (REF )
\( x^1 = x^0 \) 
\( t = 1,\cdots ,J \) 
\( \bar{x} = x^{t} + \frac{t}{t + 3} (x^{t} - x^{t - 1}) \) 
\( g = \nabla _{1} R(\bar{x}, \phi ) \) 
ever\( x^{t + 1} = \operatorname{prox}_{\alpha D(\cdot , f)}(\bar{x} - \alpha g) \) 
\( Q=R(\bar{x}, \phi ) + \langle g, x^{t + 1} - \bar{x} \rangle + \frac{1}{2 \alpha } \left\Vert x^{t + 1} - \bar{x}\right\Vert _2^2 \) 
\( R(x^{t+1}, \phi ) \le Q \)\( \alpha \leftarrow \alpha / \gamma _1 \)break
\( \alpha \leftarrow \gamma _2\alpha \) 


Accelerated proximal gradient descent with Lipschitz-backtracking.
